\section{2D Elements}
   \subsection{CTRIA3 Element}
     \[ A = \frac{1}{2} |(p_1-p_2) \times (p_1-p_3)| \]
     \[ m = (\rho +nsm) A t \]

     %verify
     \[ [J] = \left[ \begin{array}{ccc}
               1 &     1 & 1   \\
               x_1 & x_2 & x_3 \\
               y_1 & y_2 & y_3
            \end{array}\right]\]
     \[ [J]^-1 = \frac{1}{2A} \left[ \begin{array}{ccc}
               x2y3-x3y2 & y2-y3 & x3-x2 \\
               x3y1-x1y3 & y3-y1 & x3-x1 \\
               x1y2-x2y1 & y1-y2 & x2-x1
            \end{array}\right]\]
     \[ [J]^-1 = \frac{1}{2A} \left[ \begin{array}{ccc}
               2A_{23} & y_{23} & x_{32} \\
               2A_{31} & y_{31} & x_{31} \\
               2A_{12} & y_{12} & x_{21}
            \end{array}\right]\]
     \[ A_{23} = A_{31} = A{12} = \frac{A}{3} \]

     \[ [N] = \left[ \begin{array}{cccccc}
               N1 &  0 & N2 &  0 & N3 & 0  \\
               0  & N1 &  0 & N2 &  0 & N3
            \end{array}\right]\]
     \[ [B] = \frac{1}{2A} \left[ \begin{array}{cccccc}
               y_{23} &      0 & y_{31} &   0    & y_{12} &   0    \\
               0      & x_{32} &      0 & x_{13} &   0    & x_{21} \\
               x_{32} & x_{23} & x_{13} & y_{31} & x_{21} & y_{12}
            \end{array}\right]\]
     \[ e=[D] [N] u_e = [B] u_e    \]
     \[  [K_e] = |J| [B]^T [D] [B] \]

   \subsection{CTRIA6 Element}
     \[ A = \frac{1}{2} |(p_1-p_2) \times (p_1-p_3)| \]
     \[ m = (\rho +nsm) A t \]

   \subsection{CQUAD4 Element}
     \[ A = \frac{1}{2} |(p_1-p_3) \times (p_2-p_4)| \]
     \[ m = (\rho +nsm) A t \]

     %verify
    \[ [J] = \left[ \begin{array}{cccc}
               1 &     1 & 1   & 1   \\
               x_1 & x_2 & x_3 & x_4 \\
               y_1 & y_2 & y_3 & y_4 
            \end{array}\right]\]
     \[  [K_e] = |J| [B]^T [D] [B] \]

   \subsection{CQUAD8 Element}
     \[ A = \frac{1}{2} |(p_1-p_3) \times (p_2-p_4)| \]
     \[ m = (\rho +nsm) A t \]

